To find a subgroup of order 6 in a group $G$ of order 42 using Sylow's theorems, we proceed as follows:

1. Determine the prime factorization of the group order: $|G| = 42 = 2 \times 3 \times 7$

2. Identify the Sylow $p$-subgroups for each prime $p$:

   • Sylow 2-subgroups: A Sylow 2-subgroup has order $2^1 = 2$. Let $n_2$ denote the number of Sylow 2-subgroups. According to Sylow's theorems, $n_2$ divides $42/2 = 21$ and $n_2 \equiv 1 \pmod{2}$. Thus, $n_2$ can be 1 or 21.
   • Sylow 3-subgroups: A Sylow 3-subgroup has order $3^1 = 3$. Let $n_3$ denote the number of Sylow 3-subgroups. According to Sylow's theorems, $n_3$ divides $42/3 = 14$ and $n_3 \equiv 1 \pmod{3}$. Thus, $n_3$ can be 1 or 7.
   • Sylow 7-subgroups: A Sylow 7-subgroup has order $7^1 = 7$. Let $n_7$ denote the number of Sylow 7-subgroups. According to Sylow's theorems, $n_7$ divides $42/7 = 6$ and $n_7 \equiv 1 \pmod{7}$. Thus, $n_7$ can only be 1.

Since $n_7 = 1$, there is a unique Sylow 7-subgroup in $G$. Let's denote this subgroup by $P_7$. Because $P_7$ is unique, it is normal in $G$.

3. Find the intersection and product of Sylow subgroups: To form a subgroup of order 6, we need to combine a Sylow 2-subgroup and a Sylow 3-subgroup. Let's denote the Sylow 2-subgroup by $P_2$ and the Sylow 3-subgroup by $P_3$.

4. Combine Sylow 2-subgroup and Sylow 3-subgroup: If $P_2$ and $P_3$ intersect trivially, their product will have the desired order. The order of the product $P_2P_3$ is given by: $|P_2P_3| = \frac{|P_2| \cdot |P_3|}{|P_2 \cap P_3|}$ Since $P_2$ and $P_3$ intersect trivially (i.e., $|P_2 \cap P_3| = 1$), we get: $|P_2P_3| = |P_2| \cdot |P_3| = 2 \times 3 = 6$

Thus, the product of a Sylow 2-subgroup and a Sylow 3-subgroup forms a subgroup of order 6 in $G$.

Summary:

By applying Sylow's theorems, we conclude that $G$ contains subgroups of order 6, which are the products of its Sylow 2-subgroups and Sylow 3-subgroups. Specifically, these subgroups can be constructed as the product of any Sylow 2-subgroup and any Sylow 3-subgroup, assuming they intersect trivially.

Would you like further details or have any questions?

Here are some related questions you might find interesting:

1. How many Sylow 2-subgroups can exist in a group of order 42?
2. What is the significance of a normal Sylow subgroup?
3. Can a Sylow $p$-subgroup be non-normal?
4. How can we determine the number of Sylow 3-subgroups if $n_3 = 7$?
5. What are the possible structures of a group of order 42?
6. How do the Sylow theorems help in understanding the composition of a group?
7. What are the general properties of Sylow subgroups in finite groups?
8. Can there be more than one subgroup of a given order in a group of order 42?

Tip: Always check for normality of subgroups when applying Sylow theorems, as normal subgroups often simplify the structure analysis of a group.